Attitude angle sensor correcting apparatus for an artificial satellite

ABSTRACT

An attitude angle sensor correcting apparatus for an artificial satellite of the present invention includes a satellite attitude estimator. The satellite attitude estimator reads geographical image data out of an image data memory, produces a GCP (Ground Control Point) position included in the image data by stereo image measurement, and then estimates the instantaneous satellite attitude angle on the basis of a relation between the measured GCP position and a true GCP position. An attitude angle sensor data corrector corrects measured attitude angle data with estimated satellite attitude data output from the satellite attitude estimator and corresponding in time to the measured attitude angle data. The attitude angle sensor data corrector outputs an estimated satellite attitude signal.

BACKGROUND OF THE INVENTION

The present invention relates to an attitude angle sensor correctingapparatus for correcting measured attitude angle data, which is outputfrom an attitude angle sensor mounted on an artificial satellite, withan estimated attitude angle derived from geographical image data.

A conventional attitude angle sensor correcting apparatus for asatellite application includes an attitude angle sensor data memory andan attitude angle sensor noise corrector. The attitude angle sensornoise corrector produces an attitude angle correction signal by usingmeasured attitude angle data read out of the attitude angle sensor datamemory. The prerequisite with the attitude angle sensor correctingapparatus is that a positional relation between an attitude angle sensorand the center of gravity of a satellite on which it is mounted isprecisely measured and is strictly controlled even in the space. When anerror (alignment error) occurs in the attitude angle sensor due to somecause, sensor correction accuracy is critically lowered. Moreover,because a reference value for correcting alignment errors is notavailable, the detection of alignment errors itself is not practicable.

Technologies relating to the present invention are disclosed in, e.g.,Japanese Patent Laid-Open Publication Nos. 59-229667, 1-237411, 7-329897and 11-160094 as well as in Japanese Patent Publication No. 61-25600.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide anattitude angle sensor correcting apparatus capable of shootingtridimensionally a plurality of GCPs (Ground Control Points) located onthe ground with, e.g., a camera, producing estimated satellite attitudedata from measured GCP values and true GCP values, and correctingmeasured attitude angle data with the estimated attitude data to therebycorrect attitude angle sensor data.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other objects, features and advantages of the presentinvention will become more apparent from the following detaileddescription taken with the accompanying drawings in which:

FIG. 1 is a block diagram schematically showing a conventional attitudeangle sensor correcting apparatus for an artificial satellite;

FIG. 2 is a schematic block diagram showing an attitude angle sensorcorrecting apparatus embodying the present invention;

FIG. 3 is a view for describing the principle of stereo imagemeasurement to be executed by a satellite attitude estimator included inthe illustrative embodiment; and

FIG. 4 is a view demonstrating GCP correction to be also executed by thesatellite attitude estimator.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

To better understand the present invention, brief reference will be madeto a conventional attitude angle sensor correcting apparatus mounted onan artificial satellite, shown in FIG. 1. As shown, the apparatusincludes an attitude angle sensor data memory 101 and an attitude anglesensor noise corrector 102. Measured attitude angle data 103 is read outof the attitude angle sensor data memory 101. The attitude angle sensornoise corrector 102 outputs an attitude angle correction signal 104.

Specifically, the attitude angle sensor data memory 101 stores themeasured attitude angle data 103. The attitude angle sensor noisecorrector 102 estimates measurement noise contained in the attitudeangle data 103 by using a statistical probability model. The corrector102 then removes noise components from the attitude angle data 103 andoutputs the resulting data in the form of the attitude angle correctionsignal 104. With this circuitry, the apparatus corrects a measuredattitude angle sensor signal.

The correctness of the statistical probability model, or noise model,applied to the attitude angle sensor noise corrector 102 directlyeffects the accuracy of the attitude angle correction signal 104. As fora sensor noise model, various mathematically outstanding schemes haveheretofore been proposed and may be used to enhance accurate estimationto a certain degree.

The prerequisite with the attitude angle sensor correcting apparatusdescribed above is that a positional relation between an attitude anglesensor and the center of gravity of a satellite on which it is mountedis precisely measured and is strictly control led even in the space.Alignment errors critically lower the sensor correction accuracy, asstated earlier. Moreover, because a reference value for correctingalignment errors is not available, the detection of alignment errorsitself is not practicable, as also stated previously.

Referring to FIG. 2, an attitude angle sensor correcting apparatusembodying the present invention and mounted on an artificial satellitewill be described. As shown, the apparatus includes an attitude anglesensor data corrector 1, a satellite attitude estimator 2, an image datamemory 3, and an attitude angle sensor data memory 101. There are alsoshown in FIG. 2 geographical shot data 4, estimated satellite attitudedata 5, an estimated satellite attitude angle signal 6, and measuredattitude angle data 103.

The image data memory 3 stores the geographical shot data 4representative of a plurality of shots of the same geographical area onthe ground where a GCP is located. The satellite attitude estimator 2reads the data 4 out of the image data memory 3 and determines, bystereo image measurement, the measured position of the GCP contained inthe data 4. The estimator 2 then estimates the instantaneous satelliteattitude angle on the basis of a relation between the measured positionof the GCP and the true position of the same. The estimator 2 feeds theresulting estimated satellite attitude data 5 to the attitude anglesensor data corrector 1. In response, the attitude angle sensor datacorrector 1 corrects the measured attitude angle data 103 with the abovedata 5 coincident in time with the data 103 and then outputs theestimated satellite attitude angle signal 6.

The stereo image measuring method, which is a specific scheme formeasuring the positions of a plurality of GCPs located on the ground,will be described specifically hereinafter. FIG. 3 shows a specificrelation between two cameras 10 and 11 different in position from eachother and a single point P14 shot by the cameras 10 and 11. In practice,a single camera implements the two cameras 10 and 11 and shoots thesingle point P14 at different positions to thereby output two differentgeographical image data. Vectors shown in FIG. 3 derive an equation:

P₁=P_(d)+P₂  Eq. (1)

Assume that the cameras 10 and 11 have coordinates Σ_(s1) and Σ_(s2),respectively, and that the component vectors of the individual vectorsare expressed as:

 ^(S1)P₁=[^(s1)x1 ^(s1)y1 ^(s1)z1]^(T)

^(S2)P₂=[^(s2)x2 ^(s2)y2 ^(s2)z2]^(T)

Further, assume that projection points on screens 12 and 13 included inthe cameras 10 and 11, respectively, are (^(s1)x′₁ ^(s1)y′₁) and(^(s2x′) ₂ ^(s2Y′) ₂), and that the cameras 10 and 11 both have a focaldistance h. Then, there hold the following relations: $\begin{matrix}\begin{matrix}{{{{}_{}^{}{}_{}^{}} = {h{\quad \frac{{}_{}^{}{}_{}^{}}{{}_{}^{}{}_{}^{}}}}},{{{}_{}^{}{}_{}^{}} = {h{\quad \frac{{}_{}^{}{}_{}^{}}{{}_{}^{}{}_{}^{}}}}},} \\{{{{}_{}^{}{}_{}^{}} = {h{\quad \frac{{}_{}^{}{}_{}^{}}{{}_{}^{}{}_{}^{}}}}},{{{}_{}^{}{}_{}^{}} = {h{\quad \frac{{}_{}^{}{}_{}^{}}{{}_{}^{}{}_{}^{}}}}}}\end{matrix} & {{Eq}.\quad (2)}\end{matrix}$

The projection points (^(s1)x′₁ ^(s1)y′₁) and (^(s2)x′₂ ^(s2)y′₂) on thescreens 12 and 13 may alternatively be expressed as: $\begin{matrix}\begin{matrix}{{{{}_{}^{}{}_{}^{}} = {k_{x}\quad \frac{i_{1}}{\upsilon_{sx}}}},{{{}_{}^{}{}_{}^{}} = {k_{y}\quad \frac{- j_{1}}{\upsilon_{sy}}}},} \\{{{{}_{}^{}{}_{}^{}} = {k_{x}{\quad \frac{i_{2}}{\upsilon_{sx}}}}},{{{}_{}^{}{}_{}^{}} = {k_{y}\quad \frac{- j_{2}}{\upsilon_{sy}}}}}\end{matrix} & {{Eq}.\quad (3)}\end{matrix}$

where (i₁j₁) and (i₂j₂) denote pixel values corresponding to theprojection points on the screens 12 and 13, respectively, v_(sx) andv_(sy) denote a screen size, and k_(x) and k_(y) denote an image size.

Let a DCM (Direct Cosine Matrix) representative of a relation betweenthe coordinates Σ_(s1) and Σ_(s2) be expressed as:

Σ_(s2)=^(s2)C_(s1)Σ_(s1)  Eq. (4)

Then, the Eq. (1) may be rewitten as:

^(s1)P₁=^(s1)P_(d)+^(s2)C_(s1) ^(T s2)P₂  Eq. (5)

The Eqs. (2) and (5) therefore derive ^(s1)z1, as follows:$\begin{matrix}{{{}_{}^{}{}_{}^{}} = {h\quad \frac{\left( {{{{}_{}^{}{}_{}^{}}c_{3}} - h_{c1}} \right)^{T} \cdot {Pd}}{\left( {{{{}_{}^{}{}_{}^{}}c_{3}} - {hc}_{1}} \right)^{T} \cdot s}}} & {{Eq}.\quad (6)}\end{matrix}$

where

^(s2)C_(s1)=[c₁c₂c₃]^(T), s=[^(s1)x′₁ ^(s1)y′₁h]  Eq. (7)

Hereinafter will be described how the satellite attitude estimator 2generates the estimated satellite attitude data 5 by using a measuredGCP position vector ^(s)P′₁, which is derived from the image data by theEqs. (2), (3) and (6), and the true GCP position vector ^(s)P₁. FIG. 4shows the principle of satellite attitude estimation using a GCP. Asshown, assume that the vectors ^(s)P′₁ and ^(s)P₁ are respectivelyassigned to a GCP 24 in an observed image 23 and a GCP 22 in an actualimage 21. Then, the two vectors ^(s)P′₁ and ^(s)P₁ are related asfollows:

^(s)P′₁=R^(s)P₁+^(s)t  Eq. (8)

where R denotes a rotational transform matrix, i.e., RR^(T)=R^(T)R=1 anddetR=1, and ^(s)t denotes a translational transform vector.

The rotational transform matrix Rand translational transform vector^(s)t are representative of a difference between the attitudes of thecamera 20 with respect to the GCPs 22 and 24. When the camera 23 isaffixed to a satellite, the above matrix R and vector ^(s)t may directlybe interpreted as a difference in the attitude of the satellite.

Further, in an ideal condition wherein disturbance is absent, it isgenerally possible to precisely calculate the attitude of a satellitefrom time. Therefore, the true GCP position vector ^(s)P₁ indicative ofthe GCP in the actual image easily derives the attitude value of asatellite in the ideal condition. It follows that if the rotationaltransform matrix R and translational transform vector st included in theEq. (8) can be determined on the basis of the two vectors ^(s)P₁ and^(s)P′₁, there can be generated the instantaneous estimated satelliteattitude data 5.

More specifically, the satellite attitude estimator 2 first executes thestereo image measurement with the geographical shot data 4 in order toproduce a measured GCP value based on the Eqs. (2), (3) and (6). Theestimator 2 then determines a rotational transform matrix R and atranslational transform vector ^(s)t that satisfy the Eq. (8) withrespect to the measured GCP value and the true GCP value. In thismanner, the estimator 2 can generate the estimated satellite attitudedata 5 for correcting errors contained in the measured attitude angledata 103.

The estimated satellite attitude data 5 and measured attitude angle data103 are input to the attitude angle sensor data corrector 1. Theattitude angle sensor data corrector 1 detects, by using timeinformation included in the data 5, measured attitude angle data 103corresponding to the time information, compares the detected data 103with the estimated satellite attitude data 5, and then corrects the data103. In this manner, the corrector 1 corrects the data 103 on the basisof information derived from an information source that is entirelydifferent from the attitude angle sensor responsive to the data 103. Thecorrector 1 therefore successfully removes the noise components of theattitude angle sensor contained in the data 103 and corrects thealignment of the sensor, so that the estimated satellite attitude signal6 is highly accurate.

In the illustrative embodiment, the positional errors of a cameramounted on a satellite may have critical influence on the correctionaccuracy of the attitude angle sensor. In practice, however, such errorsare smoothed during stereo image measurement and influence thecorrection accuracy little. The mounting errors of the camera aretherefore substantially unquestionable in the aspect of the correctionaccuracy of the measured attitude angle data 103.

As for the satellite attitude estimator 2, the rotational transformmatrix R and translational transform vector ^(s)t that satisfy the Eq.(8) can be generated by a Moore-Penrose quasi-inverse matrix. Analternative embodiment of the present invention using this scheme willbe described hereinafter.

Assume that n GCP true vectors ^(s)P₁, FIG. 4, are present, that amatrix Q having such elements is defined as: $\begin{matrix}{Q = \left\lbrack {{\begin{bmatrix}{\,^{s}{P1}} \\1\end{bmatrix}\begin{bmatrix}{\,^{s}{P2}} \\1\end{bmatrix}}\quad {\cdots \quad\begin{bmatrix}{\,^{s}{Pn}} \\1\end{bmatrix}}} \right\rbrack} & {{Eq}.\quad (9)}\end{matrix}$

and that a matrix Q′ constituted by measured GCP vectors ^(s)P′₁ isdefined as: $\begin{matrix}{Q = \left\lbrack {{\begin{bmatrix}{{{}_{}^{}{}_{}^{}}1} \\1\end{bmatrix}\begin{bmatrix}{{{}_{}^{}{}_{}^{}}2} \\1\end{bmatrix}}\quad {\cdots \quad\begin{bmatrix}{{{}_{}^{}{}_{}^{}}n} \\1\end{bmatrix}}} \right\rbrack} & {{Eq}.\quad (10)}\end{matrix}$

Then, a simultaneous transform matrix H constituted by the matrix R andvector ^(s)t is expressed as:

H=Q′Q+  Eq. (11)

${H = \begin{bmatrix}R & {\,^{s}t} \\o & 1\end{bmatrix}},$

The Eq. (11) therefore derives a rotational transform matrix R and atranslational transform vector ^(s)t that indicate a satellite attitudeerror, which in turn derives estimated satellite data 5.

Further, the satellite attitude estimator 2 may alternatively generatethe rotational transform matrix R and translational transform vector^(s)t, which satisfy the Eq. (8), in relation to a constant coefficientmatrix. Specifically, in another alternative embodiment of the presentinvention to be described, the estimator 2 generates the above matrix Rand vector ^(s)t on the basis of the following relation.

In FIG. 4, assume that n true GCP vectors ^(s)P₁ are present, and that nmeasured GCP vectors ^(s)P′₁ corresponding thereto are present. In theembodiment to be described, the following new vectors are defined:

^(s)W_(n)=^(s)P_(n)/^(s)Z_(n)  Eq. (12)

 ^(s)W_(n)=^(s)P′_(n)/^(s)Z_(n)   Eq. (13)

Let a matrix E be defined as: $\begin{matrix}{E = {{\,^{s}\overset{\sim}{tR}} = {\begin{bmatrix}e_{1} & e_{2} & e_{3} \\e_{4} & e_{5} & e_{6} \\e_{7} & e_{8} & 1\end{bmatrix} = \begin{bmatrix}\varphi_{1}^{T} \\\varphi_{2}^{T} \\\varphi_{3}^{T}\end{bmatrix}}}} & {{Eq}.\quad (14)}\end{matrix}$

where ${\,^{s}\overset{\sim}{t}} = \begin{bmatrix}0 & {\quad^{s}t_{3}} & {- {{}_{}^{}{}_{}^{}}} \\{- {{}_{}^{}{}_{}^{}}} & 0 & {{}_{}^{}{}_{}^{}} \\{{}_{}^{}{}_{}^{}} & {- {{}_{}^{}{}_{}^{}}} & 0\end{bmatrix}$

Then, the matrix can be unconditionally determined by the followingequation:

W′(diagE).W=0  Eq. (15)

where

W=[^(s)W₁ ^(s)W₂ . . . ^(s)W_(n)],

W′=[^(s)W′₁ ^(s)W′₂ . . . ^(s)W′_(n)]

Let the matrix produced by the Eq. (15) be expressed, by singular valueresolution, as:

E=UΛV^(T)  Eq. (16)

Then, the matrix R and vector ^(s)t can eventually be determined by:$\begin{matrix}{{R = {{U\begin{bmatrix}0 & {\pm 1} & \quad \\{\pm 1} & 0 & \quad \\\quad & \quad & s\end{bmatrix}}V^{T}}},{s = {\left( {\det \quad U} \right)\left( {\det \quad V} \right)}}} & {{Eq}.\quad (17)}\end{matrix}$

$\begin{matrix}{{\,^{s}t} = {\alpha \begin{bmatrix}{\varphi_{1}^{T}{\varphi_{2}/\varphi_{2}^{T}}\varphi_{3}} \\{\varphi_{1}^{T}{\varphi_{2}/\varphi_{1}^{T}}\varphi_{3}} \\1\end{bmatrix}}} & {{Eq}.\quad (18)}\end{matrix}$

where α denotes any desired constant.

With the Eqs. (17) and (18), it is possible to determine a rotationaltransform matrix R and a translational transform vector ^(s)t indicativeof a satellite attitude error in the same manner as in the immediatelypreceding embodiment. The matrix R and vector ^(s)t derive estimatedsatellite attitude data 5, as stated earlier.

The difference between this embodiment and the immediately precedingembodiment as to the rotational transform matrix R and translationaltransform vector ^(s)t will be described more specifically by usingspecific numerical values. Assume that the following eight GCPs exist inany desired geographical data 4:

^(s)P₁₌ ^(|)[6.2 26.8 0.5 ]^(T), ^(s)P₂=[−6.5 20.1 0.08]^(T)

^(s)P₃=[7.6 −30.8 10.6]^(T), ^(s)P₄=[−0.8 −28.2 3.1]^(T)

^(s)P₅=[10.7 34.3 16.1]^(T), ^(s)P₆=[9.3 −18.6 0.15]^(T)

^(s)P⁷=[−17.2 30.1 9.5]^(T), ^(s)P₈=[16.1 24.7 2.9]^(T)  Eq. (19)

A true rotational transform matrix R and a translational transformvector ^(s)t corresponding to the above GCPs are given by:$\begin{matrix}{{R = \begin{bmatrix}0.99848 & {- 0.01562} & {- 0.05291} \\0.01742 & 0.99927 & 0.03398 \\0.05234 & {- 0.03485} & 0.99802\end{bmatrix}},{{\,^{s}t} = \begin{bmatrix}{- 81} \\31 \\{- 24}\end{bmatrix}}} & {{Eq}.\quad (20)}\end{matrix}$

Values produced by applying the transform of the Eq. (20) to the Eq.(19) and containing suitable noise are assumed to be measured GCP points^(s)P′₁, ^(s)P′₂, ^(s)P′₃, _(s)P′₄, ^(s)P′₅, ^(s)P′₆, ^(s)P′₇ and^(s)P′₈. Then, the previous embodiment produces an estimated rotationaltransform matrix R₁ and an estimated translational transform vector^(s)t₁: $\begin{matrix}{{R_{1} = \begin{bmatrix}0.99648 & {- 0.01611} & {- 0.05264} \\{0/01743} & 0.99927 & {0/03398} \\0.05276 & {- 0.03444} & {0/99568}\end{bmatrix}},{{\det \quad R_{1}} = 0.99567},{{{}_{}^{}{}_{}^{}} = \begin{bmatrix}{- 80.97565} \\31.0 \\{- 23.97823}\end{bmatrix}}} & {{Eq}.\quad (21)}\end{matrix}$

Likewise, the illustrative embodiment produces an estimated rotationaltransform matrix R and an estimated translational transform vector^(s)t₂: $\begin{matrix}{{R_{2} = \begin{bmatrix}0.99595 & {- 0.01225} & {{- 0.}\quad 08907} \\0.00772 & 0.99866 & {{- 0.}\quad 0511} \\0.08957 & 0.05020 & {0.\quad 99471}\end{bmatrix}},{{\det \quad R_{2}} = 1},{{{}_{}^{}{}_{}^{}} = \begin{bmatrix}{- 29.53341} \\38.34762 \\{- 26.40867}\end{bmatrix}}} & {{Eq}.\quad (22)}\end{matrix}$

As the Eqs. (21) and (22) indicate, the two embodiments are capable ofestimating the rotational transform matrix R and translational transformvector ^(s)t with acceptable accuracy with respect to true valuesalthough they include some errors as to numerical values.

As stated above, in accordance with the present invention, an image datamemory stores two different geographical shot data representative of thesame geographical area, where a GCP is set, shot at two differentpoints. A satellite attitude estimator reads the shot data out of theimage data memory, determines the position of the GCP included in theimage data by stereo image measurement, and estimates the instantaneousattitude angle of a satellite by referencing a relation between thedetermined GCP position and a true GCP position. The estimated satelliteattitude angle is input to an attitude angle sensor data corrector asestimated satellite attitude data. In response, the attitude anglesensor data corrector corrects measured attitude angle data output froman attitude angle sensor data memory with the estimated satelliteattitude data corresponding in time to the measured data.

The attitude angle sensor may be implemented by the integrated value ofa gyro signal, STT or an earth sensor by way of example. Also, the imagedata can be easily attained with a camera mounted on a satellite.

In summary, the present invention provides an attitude angle sensorcorrecting apparatus for an artificial satellite having the followingunprecedented advantages. The correcting apparatus includes an imagedata memory, an attitude angle estimator, and an attitude angle sensordata corrector. The correcting apparatus can therefore remove both ofsensing errors ascribable to random noise and bias noise, which areparticular to an attitude angle sensor, and the alignment errors of thesensor mounted on a satellite, thereby determining an attitude anglewith utmost accuracy. The correcting apparatus further promotes theaccurate determination of an attitude angle by producing measured GCPvalues by executing stereo image measurement with geographical shotdata.

Various modifications will become possible for those skilled in the artafter receiving the teachings of the present disclosure withoutdeparting from the scope thereof.

What is claimed is:
 1. An attitude angle correcting apparatus for anartificial satellite, comprising: an attitude angle sensor data memoryfor storing a signal output from sensing means responsive to an attitudeangle of the artificial satellite; an image data memory for storinggeographical image data representative a same geographical area, where a(Ground Control Point) GCP is located, shot at a plurality of positions;a satellite attitude estimator for generating estimated attitude data ofthe artificial satellite on the basis of a difference between a true GCPvalue representative of a true position of the GCP and a measured GCPvalue produced by image measurement using said geographical image datastored in said image data memory; and an attitude angle sensor datacorrector for estimating an attitude angle of the artificial satelliteby using said estimated attitude data, and then correcting measuredattitude angle data, which is read out of said attitude angle sensordata memory, with said attitude angle estimated to thereby generate anestimated attitude angle signal.
 2. An apparatus as claimed in claim 1,wherein a plurality of GCPs (Ground Control Points) are located ongeometry represented by said geometrical image data.
 3. An apparatus asclaimed in claim 2, wherein said satellite attitude estimator estimatesan attitude angle of the artificial satellite by describing, for each ofthe GCPs whose positions can be measured on the basis of saidgeographical image data and whose true values are known, a relationbetween the measured GCP value and the true GCP value by use of aMoor-Penrose quasi-inverse matrix.
 4. An apparatus as claimed in claim2, wherein when said satellite attitude estimator relates the measuredvalue and the true value of each of the GCPs, positions of which can bemeasured on the basis of the geographical image data and true values ofwhich are known, by using a constant coefficient matrix:$E = \begin{bmatrix}e_{1} & e_{2} & e_{3} \\e_{4} & e_{5} & e_{6} \\e_{7} & e_{8} & 1\end{bmatrix}$

said satellite attitude estimator estimates an attitude angle error ofthe artificial satellite by using a result of singular value resolutionof said constant coefficient matrix.
 5. An apparatus as claimed in claim1, wherein said satellite attitude estimator estimates an attitude angleof the artificial satellite by describing, for each of the GCPs whosepositions can be measured on the basis of said geographical image dataand whose true values are known, a relation between the measured GCPvalue and the true GCP value by use of a Moor-Penrose quasi-inversematrix.
 6. An apparatus as claimed in claim 1, An apparatus as claimedin claim 2, wherein when said satellite attitude estimator relates themeasured value and the true value of each of the GCPs, positions ofwhich can be measured on the basis of the geographical image data andtrue values of which are known, by using a constant coefficient matrix:$E = \begin{bmatrix}e_{1} & e_{2} & e_{3} \\e_{4} & e_{5} & e_{6} \\e_{7} & e_{8} & 1\end{bmatrix}$

said satellite attitude estimator estimates an attitude angle error ofthe artificial satellite by using a result of singular value resolutionof said constant coefficient matrix.